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Vector bundle

Geometric Academic Language

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A vector bundle is a geometric structure. For each point of a topological space (or manifold, or algebraic cluster), a vector space is attached in a mutually compatible way. These vector spaces used "stick together" to form a new topological space (or manifold, or algebraic cluster).^[1]

A typical example is the tangent bundle of a manifold: attach the Tangent space 。 Or consider a plane Smooth curve , then attach a straight line perpendicular to the curve at each point of the curve; This is the curve“ Normal bundle "。

The vector bundle is Fibre plexus One of.

Chinese name: Vector bundle

Foreign name: vector bundle
Discipline: Fiber bundle theory

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seven Operation of vector bundle
eight Variants and promotion

definition

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Vector bundle π: E → M is fibre by vector space Of Fibre plexus 。^[4]

Related concepts

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Let ξ=π: E → M be a vector bundle.

ξ's Euclid metric Is (ξ ⊗ ξ) * section s， Satisfy that any b, s (b) in M is fiber E b Of inner product 。

Any vector bundle is allowed to have Euclidean metric.

Of topological space M Cutterplex Euclidean measure of TM, called M Riemann metric , and said that M has Riemannian Structure , which defines the manifold be called Riemannian manifold 。^[3]

The vector bundle is a parallelizable manifold if it is a trivial bundle.^[5]

Operation of plex

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The complex conjugate of complex vector bundle η is

。

The fiber space of complex c (η) of real vector bundle η is ℂ ⊗ R F， The structure group is G × G.

Complex vector bundle η one Materialized r (η) of one ）Meet cr (η one )=η⨁

，rc(η)=η⨁η。^[6]

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Let ξ be n-dimensional Smooth manifold Vector bundles of order n on B. B can be n+1 set U zero ,..., U n Overlay, so as to limit ξ| Ui It is an ordinary bundle.

Let ξ be the vector bundle of order n on B, γ n,k by Glassman manifold G n,k Order n on There is a myriad of clumps When l is large enough, there is a mapping f: B → G n,l , satisfying ξ ≅ f * γ n,l 。 G n,l be called Classification space , f is called Classification mapping 。

If B is a k-dimensional smooth flow, then there is a bijection Vect n (B) ↔ [B,G n,n(2k+1) ], where Vect n (B) Is the equivalent class of vector bundle ξ on B, [B, G n,n(2k+1) ]Is the homotopy class of ∘ f, f: B → G n,k Is the classification mapping of ξ, I: G n,nk →G n,n(2k+1) Include mapping for.^[5]

vector bundle morphism

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A Slave Vector Bundle π one : E one → X one To vector bundle π two : E two → X two Of Morphisms （morphism） Is a pair of continuous maps f : E one → E two and g : X one → X two bring^[2]

g π one = π two f
For each X one In x , by f Induced mapping π one ({ x }) → π two ({ g ( x ）}) is a vector space linear transformation 。

All classes of vector bundles and projections of bundles form a category 。 If we limit it to smooth manifolds and smooth bundles, we have the category of smooth vector bundles.

We can consider a fixed bottom space X A category consisting of all vector bundles of. Let's take those in the bottom space X On Identity mapping (identity map) as a radio in this category That is to say, the bundle meets the following commutative graph:

vector bundle morphism

(Note that this category is not commutative; projective of vector bundles nucleus Usually it cannot be a vector bundle naturally.)

section

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Given a vector bundle π: E → X And an open subset U , we can consider that π U On section , that is, continuous function s : U → E Meet π s = id U . In essence, the section gives U A vector taken from the vector space attached to the point at each point of, and the value should be continuous.

For example, the section of the tangent bundle of a differential manifold is the vector field on the manifold.

order F ( U ）For U Collection of all sections on the F ( U ）There is always at least one element: V In x Map to π ({ x }) s . Use addition and multiplication of each point, F ( U ）The sum of these vector spaces is X Of vector space on layer 。

if s belong to F ( U ）And α: U → R Is a continuous mapping, then α s belong to F ( U ）We can see F ( U ）Is a U On the ring of continuous real valued functions over model 。 Further, if O X express X The layer structure of upper continuous function, then F Yes O X -A layer of mold.

Not O X -Each layer of the module is derived from the vector bundle in this way: only the local free layer can be obtained from this method. (Reason: local, we need to find a projection U × R → U These are just continuous functions U → R , and this function is continuous U → R n -Tuple.)

Further speaking: X The category of real vector bundles on is equivalence On O X -Modules are locally free and finitely generated by layers.

So we can regard the vector bundle as being located at O X -In the category of module layer; The latter is commutative, so we can calculate the projective kernel of the vector bundle.

Operation of vector bundle

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Two X A vector bundle on the same field on a Whitney and The fibers at each point are those of the two bundles direct product 。 Similarly, fiber Vector product And dual space bundles can also be introduced in this way.

Variants and promotion

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Vector bundle is a special case of fiber bundle.

Smooth vector bundle Defined as meeting E and X Is a smooth manifold, π: E → X Is a smooth mapping, while the local trivial mapping φ is Differential homeomorphism The vector bundle of.

hold Real vector space If you change it to complex, you get complex vector bundles. This is a special case of reduction of structure group. You can also use other Topological domain Vector space, but relatively rare.^[1]

If we allow the use of arbitrary Banach space (not just R ）, you can get Banach cluster.

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